Points d’évaluation pour les opérateurs cycliques ayant la propriété de Bishop (β)

Let T be a linear bounded cyclic operator in a separable complex Hilbert space H. Let B(T) and Ba(T) denote, respectively, the set of bounded point evaluation and the set of analytic point evaluation of T. We show that if T has the Bishop property (β), then Ba(T)=B(T)⧹σap(T), where σap(T) is the approximate spectrum of T. In the particular case when T is an operator of multiplication by z in a Hardy space this was proved by Trent (Pacific J. Math. 80 (1979) 279). On the other hand, using the generalized and the local spectral theory we obtain sufficient conditions on Ba(T) under which the spectrum of T and the local spectrum of T at any y≠0 in H coincide. At the end results involving the spectral picture of quasi-similar cyclic operators are given.